Question 1191939: 30. A hill slopes at an angle of 12^ 28^ prime with the horizontal . From the base of the hill , the angle of inclination of a 459.0 ft tower at the top of the hill is 35^ 9^ prime . How much rope would be required to reach from the top of the tower to the bottom of the hill?
Found 2 solutions by Alan3354, math_tutor2020:
Answer by Alan3354(69443)   (Show Source):
Answer by math_tutor2020(3816)  (Show Source):
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If I had to guess, the notation 12^ 28^ prime is referring to 12°28'
The small circle is the degree symbol and not the number zero, and not the letter 'o'.
If my assumption is correct, then that means the angle of elevation of the hill is 12 full degrees then plus another 28 arc minutes.
There are 60 arc minutes in one degree.
Think of a round analog clock how it is split into 60 equal slices.
Therefore, saying 28 arc minutes means we add on 28/60 of a full degree to the 12 full degrees already there.
In other words,
12°28' = 12 + 28/60
12°28' = 12.4667° approximately
Similarly,
35°9' = 35 + 9/60
35°9' = 35.15° exactly
This is what the diagram would look like
A = base of the hill
B = base of the tower
C = top of the tower
The rope connects A to C. The length of which is x.
The goal of course is to find x.
Let's introduce a fourth point D.
I'll put it on the horizontal line directly below point B.
Focus on triangle ACD
This is a right triangle
The lower acute angle is A = 35.15
The upper acute angle is C = 90 - A = 90-35.15 = 54.85
Now focus on triangle ABC
The difference of the blue and green angles is 35.15 - 12.4667 = 22.6833 approximately.
This is the interior angle A of triangle ABC.
For any triangle, the three interior angles always add to 180.
A+B+C = 180
22.6833 + B + 54.85 = 180
B + 77.5333 = 180
B = 102.4667
This is approximate.
Now we can use the law of sines to find side b of triangle ABC. Side b is opposite angle B.
Only focus on triangle ABC
sin(A)/a = sin(B)/b
sin(22.6833)/459 = sin(102.4667)/x
0.00084017 = 0.97642164/x
0.00084017x = 0.97642164
x = 0.97642164/0.00084017
x = 1162.17151291
x = 1162.17
Answer: Approximately 1162.17 feet of rope is needed
This value is approximate because the other values mentioned were approximate.
Use more decimal digits in the steps to get a more accurate answer.
Also keep in mind that realistically we'd need a bit more rope at each end so that each end can be tied down. So if your teacher wanted things to the nearest foot, then its a good idea to round up to 1163 feet even though 1162.17 is closer to 1162
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